5-adic integers - Knowlpedia

The $5$ -adic integer ring is the closed unit ball

\mathbb Z_5=\{x\in\mathbb Q_5:|x|_5\le1\}

in $\mathbb Q_5$ , with inherited addition, multiplication and $5$ -adic topology. Equivalently it is the inverse limit $\varprojlim_m\mathbb Z/5^m\mathbb Z$ under reduction maps.

Digits and residue

Every element has a unique convergent expansion $\sum_{j\ge0}a_j5^j$ , with digits $a_j\in\{0,\ldots,5-1\}$ . The constant digit is reduction modulo the maximal ideal $5\mathbb Z_5$ ; the residue field is $\mathbb F_5$ .

Ring rather than field

The unit group consists of elements with nonzero constant digit. The ring is compact, complete and infinite, and $5$ is not a unit. In particular it is neither the finite ring $\mathbb Z/5\mathbb Z$ nor a rational vector space.

References

J. S. Milne, Algebraic Number Theory, Chapter 7, Theorem 7.23; Propositions 7.26, 7.46; Remark 7.49, pp. 115–127.